1. (10 points) Let P, Q and S be points in ℝ^3 which are not collinear. Define a subset D of ℝ^3

Step 1: The set D is the set of all points that can be written in the form $aoverrightarrow{OP}

1. (10 points) Let P, Q and S be points in ℝ^3 which are not...

1. (10 points) Let $P$, $Q$ and $S$ be points in $\mathbb{R}^3$ which are not collinear. Define a subset $D$ of $\mathbb{R}^3$ which is the set of tip of vectors $a \overrightarrow{OP} + b \overrightarrow{OQ} + c \overrightarrow{OS}$ with $a+b+c=1$. Here, $O$ is the origin. State what the set $D$ is with full explanations.

Transcript

0:00 We're calculating some volumes.

00:03 And so first of all, we're given these points.

00:08 A equals 1, 2, 3, b is 136, c is 386, d is 373, and p is minus 210, q is 232, r is 1 .4 .4 .5.

00:34 Minus 1, and s is 3, 6 ,1.

00:41 So first of all, we're asked, what's the volume of a parallel pipette determined by vectors xy, x, v, and x, z, and these are all lying in a plane? well, if they're all lying in a plane, then the volume of the shape that they form is just going to be zero is it's the volume of a flat shape.

01:14 So in plane means volume equals zero.

01:23 So first of all, we want to find out do a, b, c, d lie in a plane? and to work this out, we are going to work out whether the parallel of ipad with, that's being defined by ab, ac, and ad, we're going to find out whether this has volume zero.

01:59 If it has volume zero, then these points lie in a plane.

02:05 So a, b is going to be b minus a.

02:15 So b minus a is 1 minus 1, 0, 3 minus 2 is 1.

02:20 So 0, 1, so 0, 1, 3, 3, 3.

02:22 And similarly we've got ac and this is going to be 263 and ad is going to be 2 -6 -3 and ad is going to be 2 -50.

02:46 And the volume of the shape that they're forming, v is going to be the absolute value of the scalar triple product.

03:07 Ab dot ac sorry ab cross ac dot ad so it's the absolute value so let's do ab cross ac so let's write this out so n -1 -3 crossed with 263 and then that dotted with 250 and we take the absolute value of that so first of all, we do this cross product.

03:50 So this is going to be 3 minus 18 is minus 15.

03:57 And then minus.

04:00 So, and then this is going to be 6 minus 0.

04:06 There's 6.

04:07 And then the bottom one is 0 minus 2.

04:12 So minus 2 dotted with 250.

04:15 0.

04:19 So this is minus 30 plus 30 plus 0.

04:28 So this is 0.

04:29 So these vectors are all lying in a plane because the volume of the parallel of piped that they're creating is 0.

04:37 So a, b, c, d do lie in a plane of r3.

04:51 So now we want to work out the area of the parallelogram.